Optimal. Leaf size=45 \[ \frac {b (c d-b e)}{c^3 (b+c x)}+\frac {(c d-2 b e) \log (b+c x)}{c^3}+\frac {e x}{c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ \frac {b (c d-b e)}{c^3 (b+c x)}+\frac {(c d-2 b e) \log (b+c x)}{c^3}+\frac {e x}{c^2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {e}{c^2}+\frac {b (-c d+b e)}{c^2 (b+c x)^2}+\frac {c d-2 b e}{c^2 (b+c x)}\right ) \, dx\\ &=\frac {e x}{c^2}+\frac {b (c d-b e)}{c^3 (b+c x)}+\frac {(c d-2 b e) \log (b+c x)}{c^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.91 \[ \frac {\frac {b (c d-b e)}{b+c x}+(c d-2 b e) \log (b+c x)+c e x}{c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 69, normalized size = 1.53 \[ \frac {c^{2} e x^{2} + b c e x + b c d - b^{2} e + {\left (b c d - 2 \, b^{2} e + {\left (c^{2} d - 2 \, b c e\right )} x\right )} \log \left (c x + b\right )}{c^{4} x + b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 51, normalized size = 1.13 \[ \frac {x e}{c^{2}} + \frac {{\left (c d - 2 \, b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{3}} + \frac {b c d - b^{2} e}{{\left (c x + b\right )} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 61, normalized size = 1.36 \[ -\frac {b^{2} e}{\left (c x +b \right ) c^{3}}+\frac {b d}{\left (c x +b \right ) c^{2}}-\frac {2 b e \ln \left (c x +b \right )}{c^{3}}+\frac {d \ln \left (c x +b \right )}{c^{2}}+\frac {e x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 50, normalized size = 1.11 \[ \frac {b c d - b^{2} e}{c^{4} x + b c^{3}} + \frac {e x}{c^{2}} + \frac {{\left (c d - 2 \, b e\right )} \log \left (c x + b\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 56, normalized size = 1.24 \[ \frac {e\,x}{c^2}-\frac {b^2\,e-b\,c\,d}{c\,\left (x\,c^3+b\,c^2\right )}-\frac {\ln \left (b+c\,x\right )\,\left (2\,b\,e-c\,d\right )}{c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 44, normalized size = 0.98 \[ \frac {- b^{2} e + b c d}{b c^{3} + c^{4} x} + \frac {e x}{c^{2}} - \frac {\left (2 b e - c d\right ) \log {\left (b + c x \right )}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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